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Explore the properties and applications of 45-45-90 and 30-60-90 triangles, including side relationships and angle bisectors. Understand the concepts through examples and proofs. Enhance your geometry skills with these fundamental triangle types.
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Objectives • Justify and apply properties of 45°-45°-90° triangles. • Justify and apply properties of 30°- 60°- 90° triangles.
45-45-90 Triangles D C l A B l h2 = l2 + l2 (Pythagorean Thm) h2 = 2l2 h = D 45° h l 90° 45° A B l In a 45°-45°-90° triangle, both legs are congruent and the length of the hypotenuse is times the length of a leg. h = l
45-45-90 Example Find x.
30-60-90 Triangles (Part 1) 60° 2s 2s 30° 30° 60° 60° 60° 60° E 2s G E H G F F Given: FH bisects ∠EFG. Prove: EH = GH ∠E ≅ ∠G EF ≅ GF FH bisects ∠EFG ∠EFH ≅∠GFH ΔFEH ≅ ΔFGH EH ≅ GH Equilateral Δs are equiangular Definition of equilateral Given Defn of ∠ bisector ASA CPCTC
30-60-90 Triangles (Part 2) F 30° 30° 30° 2s 2s 2s l 60° 90° 60° 60° E H G s s s In a 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is times the length of the shorter leg. hypotenuse = 2 * shorter leg longer leg = * shorter leg l2 = (2s)2 – s2 l2 = 4s2 – s2 l2 = 3s2 l = s
